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Theorem isobs 18511
Description: The predicate "is an orthonormal basis" (over a pre-Hilbert space). (Contributed by Mario Carneiro, 23-Oct-2015.)
Hypotheses
Ref Expression
isobs.v  |-  V  =  ( Base `  W
)
isobs.h  |-  .,  =  ( .i `  W )
isobs.f  |-  F  =  (Scalar `  W )
isobs.u  |-  .1.  =  ( 1r `  F )
isobs.z  |-  .0.  =  ( 0g `  F )
isobs.o  |-  ._|_  =  ( ocv `  W )
isobs.y  |-  Y  =  ( 0g `  W
)
Assertion
Ref Expression
isobs  |-  ( B  e.  (OBasis `  W
)  <->  ( W  e. 
PreHil  /\  B  C_  V  /\  ( A. x  e.  B  A. y  e.  B  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  B
)  =  { Y } ) ) )
Distinct variable groups:    x, y,  .,    x,  .0. , y    x,  .1. , y    x, B, y   
x, W, y
Allowed substitution hints:    F( x, y)    ._|_ ( x, y)    V( x, y)    Y( x, y)

Proof of Theorem isobs
Dummy variables  h  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-obs 18496 . . . . 5  |- OBasis  =  ( h  e.  PreHil  |->  { b  e.  ~P ( Base `  h )  |  ( A. x  e.  b 
A. y  e.  b  ( x ( .i
`  h ) y )  =  if ( x  =  y ,  ( 1r `  (Scalar `  h ) ) ,  ( 0g `  (Scalar `  h ) ) )  /\  ( ( ocv `  h ) `  b
)  =  { ( 0g `  h ) } ) } )
21dmmptss 5494 . . . 4  |-  dom OBasis  C_  PreHil
3 elfvdm 5883 . . . 4  |-  ( B  e.  (OBasis `  W
)  ->  W  e.  dom OBasis )
42, 3sseldi 3495 . . 3  |-  ( B  e.  (OBasis `  W
)  ->  W  e.  PreHil )
5 fveq2 5857 . . . . . . . . 9  |-  ( h  =  W  ->  ( Base `  h )  =  ( Base `  W
) )
6 isobs.v . . . . . . . . 9  |-  V  =  ( Base `  W
)
75, 6syl6eqr 2519 . . . . . . . 8  |-  ( h  =  W  ->  ( Base `  h )  =  V )
87pweqd 4008 . . . . . . 7  |-  ( h  =  W  ->  ~P ( Base `  h )  =  ~P V )
9 fveq2 5857 . . . . . . . . . . . 12  |-  ( h  =  W  ->  ( .i `  h )  =  ( .i `  W
) )
10 isobs.h . . . . . . . . . . . 12  |-  .,  =  ( .i `  W )
119, 10syl6eqr 2519 . . . . . . . . . . 11  |-  ( h  =  W  ->  ( .i `  h )  = 
.,  )
1211oveqd 6292 . . . . . . . . . 10  |-  ( h  =  W  ->  (
x ( .i `  h ) y )  =  ( x  .,  y ) )
13 fveq2 5857 . . . . . . . . . . . . . 14  |-  ( h  =  W  ->  (Scalar `  h )  =  (Scalar `  W ) )
14 isobs.f . . . . . . . . . . . . . 14  |-  F  =  (Scalar `  W )
1513, 14syl6eqr 2519 . . . . . . . . . . . . 13  |-  ( h  =  W  ->  (Scalar `  h )  =  F )
1615fveq2d 5861 . . . . . . . . . . . 12  |-  ( h  =  W  ->  ( 1r `  (Scalar `  h
) )  =  ( 1r `  F ) )
17 isobs.u . . . . . . . . . . . 12  |-  .1.  =  ( 1r `  F )
1816, 17syl6eqr 2519 . . . . . . . . . . 11  |-  ( h  =  W  ->  ( 1r `  (Scalar `  h
) )  =  .1.  )
1915fveq2d 5861 . . . . . . . . . . . 12  |-  ( h  =  W  ->  ( 0g `  (Scalar `  h
) )  =  ( 0g `  F ) )
20 isobs.z . . . . . . . . . . . 12  |-  .0.  =  ( 0g `  F )
2119, 20syl6eqr 2519 . . . . . . . . . . 11  |-  ( h  =  W  ->  ( 0g `  (Scalar `  h
) )  =  .0.  )
2218, 21ifeq12d 3952 . . . . . . . . . 10  |-  ( h  =  W  ->  if ( x  =  y ,  ( 1r `  (Scalar `  h ) ) ,  ( 0g `  (Scalar `  h ) ) )  =  if ( x  =  y ,  .1.  ,  .0.  )
)
2312, 22eqeq12d 2482 . . . . . . . . 9  |-  ( h  =  W  ->  (
( x ( .i
`  h ) y )  =  if ( x  =  y ,  ( 1r `  (Scalar `  h ) ) ,  ( 0g `  (Scalar `  h ) ) )  <-> 
( x  .,  y
)  =  if ( x  =  y ,  .1.  ,  .0.  )
) )
24232ralbidv 2901 . . . . . . . 8  |-  ( h  =  W  ->  ( A. x  e.  b  A. y  e.  b 
( x ( .i
`  h ) y )  =  if ( x  =  y ,  ( 1r `  (Scalar `  h ) ) ,  ( 0g `  (Scalar `  h ) ) )  <->  A. x  e.  b  A. y  e.  b 
( x  .,  y
)  =  if ( x  =  y ,  .1.  ,  .0.  )
) )
25 fveq2 5857 . . . . . . . . . . 11  |-  ( h  =  W  ->  ( ocv `  h )  =  ( ocv `  W
) )
26 isobs.o . . . . . . . . . . 11  |-  ._|_  =  ( ocv `  W )
2725, 26syl6eqr 2519 . . . . . . . . . 10  |-  ( h  =  W  ->  ( ocv `  h )  = 
._|_  )
2827fveq1d 5859 . . . . . . . . 9  |-  ( h  =  W  ->  (
( ocv `  h
) `  b )  =  (  ._|_  `  b
) )
29 fveq2 5857 . . . . . . . . . . 11  |-  ( h  =  W  ->  ( 0g `  h )  =  ( 0g `  W
) )
30 isobs.y . . . . . . . . . . 11  |-  Y  =  ( 0g `  W
)
3129, 30syl6eqr 2519 . . . . . . . . . 10  |-  ( h  =  W  ->  ( 0g `  h )  =  Y )
3231sneqd 4032 . . . . . . . . 9  |-  ( h  =  W  ->  { ( 0g `  h ) }  =  { Y } )
3328, 32eqeq12d 2482 . . . . . . . 8  |-  ( h  =  W  ->  (
( ( ocv `  h
) `  b )  =  { ( 0g `  h ) }  <->  (  ._|_  `  b )  =  { Y } ) )
3424, 33anbi12d 710 . . . . . . 7  |-  ( h  =  W  ->  (
( A. x  e.  b  A. y  e.  b  ( x ( .i `  h ) y )  =  if ( x  =  y ,  ( 1r `  (Scalar `  h ) ) ,  ( 0g `  (Scalar `  h ) ) )  /\  ( ( ocv `  h ) `
 b )  =  { ( 0g `  h ) } )  <-> 
( A. x  e.  b  A. y  e.  b  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  b
)  =  { Y } ) ) )
358, 34rabeqbidv 3101 . . . . . 6  |-  ( h  =  W  ->  { b  e.  ~P ( Base `  h )  |  ( A. x  e.  b 
A. y  e.  b  ( x ( .i
`  h ) y )  =  if ( x  =  y ,  ( 1r `  (Scalar `  h ) ) ,  ( 0g `  (Scalar `  h ) ) )  /\  ( ( ocv `  h ) `  b
)  =  { ( 0g `  h ) } ) }  =  { b  e.  ~P V  |  ( A. x  e.  b  A. y  e.  b  (
x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  b )  =  { Y } ) } )
36 fvex 5867 . . . . . . . . 9  |-  ( Base `  W )  e.  _V
376, 36eqeltri 2544 . . . . . . . 8  |-  V  e. 
_V
3837pwex 4623 . . . . . . 7  |-  ~P V  e.  _V
3938rabex 4591 . . . . . 6  |-  { b  e.  ~P V  | 
( A. x  e.  b  A. y  e.  b  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  b
)  =  { Y } ) }  e.  _V
4035, 1, 39fvmpt 5941 . . . . 5  |-  ( W  e.  PreHil  ->  (OBasis `  W )  =  { b  e.  ~P V  |  ( A. x  e.  b  A. y  e.  b  (
x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  b )  =  { Y } ) } )
4140eleq2d 2530 . . . 4  |-  ( W  e.  PreHil  ->  ( B  e.  (OBasis `  W )  <->  B  e.  { b  e. 
~P V  |  ( A. x  e.  b 
A. y  e.  b  ( x  .,  y
)  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  b
)  =  { Y } ) } ) )
42 raleq 3051 . . . . . . . 8  |-  ( b  =  B  ->  ( A. y  e.  b 
( x  .,  y
)  =  if ( x  =  y ,  .1.  ,  .0.  )  <->  A. y  e.  B  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  ) ) )
4342raleqbi1dv 3059 . . . . . . 7  |-  ( b  =  B  ->  ( A. x  e.  b  A. y  e.  b 
( x  .,  y
)  =  if ( x  =  y ,  .1.  ,  .0.  )  <->  A. x  e.  B  A. y  e.  B  (
x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  ) ) )
44 fveq2 5857 . . . . . . . 8  |-  ( b  =  B  ->  (  ._|_  `  b )  =  (  ._|_  `  B ) )
4544eqeq1d 2462 . . . . . . 7  |-  ( b  =  B  ->  (
(  ._|_  `  b )  =  { Y }  <->  (  ._|_  `  B )  =  { Y } ) )
4643, 45anbi12d 710 . . . . . 6  |-  ( b  =  B  ->  (
( A. x  e.  b  A. y  e.  b  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  b
)  =  { Y } )  <->  ( A. x  e.  B  A. y  e.  B  (
x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  B )  =  { Y } ) ) )
4746elrab 3254 . . . . 5  |-  ( B  e.  { b  e. 
~P V  |  ( A. x  e.  b 
A. y  e.  b  ( x  .,  y
)  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  b
)  =  { Y } ) }  <->  ( B  e.  ~P V  /\  ( A. x  e.  B  A. y  e.  B  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  B )  =  { Y } ) ) )
4837elpw2 4604 . . . . . 6  |-  ( B  e.  ~P V  <->  B  C_  V
)
4948anbi1i 695 . . . . 5  |-  ( ( B  e.  ~P V  /\  ( A. x  e.  B  A. y  e.  B  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  B
)  =  { Y } ) )  <->  ( B  C_  V  /\  ( A. x  e.  B  A. y  e.  B  (
x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  B )  =  { Y } ) ) )
5047, 49bitri 249 . . . 4  |-  ( B  e.  { b  e. 
~P V  |  ( A. x  e.  b 
A. y  e.  b  ( x  .,  y
)  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  b
)  =  { Y } ) }  <->  ( B  C_  V  /\  ( A. x  e.  B  A. y  e.  B  (
x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  B )  =  { Y } ) ) )
5141, 50syl6bb 261 . . 3  |-  ( W  e.  PreHil  ->  ( B  e.  (OBasis `  W )  <->  ( B  C_  V  /\  ( A. x  e.  B  A. y  e.  B  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  B )  =  { Y } ) ) ) )
524, 51biadan2 642 . 2  |-  ( B  e.  (OBasis `  W
)  <->  ( W  e. 
PreHil  /\  ( B  C_  V  /\  ( A. x  e.  B  A. y  e.  B  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  B
)  =  { Y } ) ) ) )
53 3anass 972 . 2  |-  ( ( W  e.  PreHil  /\  B  C_  V  /\  ( A. x  e.  B  A. y  e.  B  (
x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  B )  =  { Y } ) )  <->  ( W  e. 
PreHil  /\  ( B  C_  V  /\  ( A. x  e.  B  A. y  e.  B  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  B
)  =  { Y } ) ) ) )
5452, 53bitr4i 252 1  |-  ( B  e.  (OBasis `  W
)  <->  ( W  e. 
PreHil  /\  B  C_  V  /\  ( A. x  e.  B  A. y  e.  B  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  B
)  =  { Y } ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807   {crab 2811   _Vcvv 3106    C_ wss 3469   ifcif 3932   ~Pcpw 4003   {csn 4020   dom cdm 4992   ` cfv 5579  (class class class)co 6275   Basecbs 14479  Scalarcsca 14547   .icip 14549   0gc0g 14684   1rcur 16936   PreHilcphl 18419   ocvcocv 18451  OBasiscobs 18493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fv 5587  df-ov 6278  df-obs 18496
This theorem is referenced by:  obsip  18512  obsrcl  18514  obsss  18515  obsocv  18517
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